Given a compound waveform, it is desirable to accurately measure the waveform and its components, which may have been spawned by several sources. This is difficult when the waveform includes signals produced by different sources overlapping in time and frequency, low energy signals eclipsed by higher energy signals, rapid changes in frequency, and/or rapid changes in amplitude. If these waveforms could be more accurately measured and analyzed, it would greatly increase our ability to understand what these waveforms contain and how to separate and/or modify them.
Analysis of waveforms is traditionally accomplished in the time and frequency domains. Typically, these waveforms are first captured digitally as amplitude samples in time, then a series of transforms are used to measure the signals and the result is displayed in a time, frequency and amplitude matrix. A variety of techniques have been developed to extract time/frequency/amplitude information from the time-series data. However, representing how the frequency and amplitude change with respect to time can be challenging, particularly when there are abrupt frequency and/or amplitude changes, or signals from multiple sources occupy the same time and frequency regions.
One common transform for obtaining time, frequency, and amplitude information is the Discrete Fourier Transform (DFT). Unfortunately, there is a tradeoff between frequency and time resolutions resulting from the size (dimension) of the DFT. The time window inspected by a DFT is proportional to its dimension. Thus, a large dimension DFT inspects a larger time window than a small dimension DFT. This larger time window makes a large dimension DFT slow to react to dynamic changes.
Conversely, a large dimension DFT slices up the frequency range into finer segments. The maximum frequency measured by the DFT is half the sampling rate of the digitized signal. A DFT of dimension X divides the frequency range from 0 to the max into X/2 equal sized “bins.” Thus, the size of each frequency bin in a DFT is equal to the sampling rate divided by its dimension.
So, higher dimension DFTs have higher frequency resolution but lower time resolution. Lower dimension DFTs have higher time resolution but lower frequency resolution. Because of this tradeoff, practitioners have sought modified DFTs or other alternative methods to accurately represent dynamic, time-varying waveforms with good resolution in both time and frequency. The Precision Measuring Matrix (PMM) (US Patent application No. PCT/US2009/064120) is just such a method. This invention extends the PMM method.
Fast Fourier Transforms (FFTs) are typically used to perform convolutions and deconvolutions. The Convolution Theorem states that under suitable conditions the Fourier transform of a convolution of two functions is the pointwise product of the Fourier transforms of the functions. It is usually simpler (and faster) to transform two functions, multiply them, and transform back than to perform a brute force convolution. More to the point, it is dramatically simpler and faster to use the convolution theorem when deconvolving. Brute force deconvolution involves solving N equations in N unknowns. Transforming the two functions, dividing, and transforming back is much easier. Brute force deconvolution is not only computationally daunting, it can be impossible. When deconvolving a series of numbers, the result is often unstable. Thus, deconvolution is, in practice, virtually always done using transforms.
The inventors have been issued several patents, which are hereby incorporated by reference. They are: Fast Find Fundamental Method, U.S. Pat. No. 6,766,288 B1; Method of Modifying Harmonic Content of a Complex Waveform, U.S. Pat. No. 7,003,120 B1; and Method of Signal Shredding, U.S. Pat. No. 6,798,886 B1. Application No. PCT/US2009/064120 filed Nov. 12, 2009 is also hereby incorporated by reference.